The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations

• Main Authors: Song He, Zhenjie Li, Chi Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2021-03-01
• Series: Journal of High Energy Physics
• Subjects: Scattering Amplitudes; Anomalies in Field and String Theories; Supersymmetric Gauge Theory
• Online Access: https://doi.org/10.1007/JHEP03(2021)278

Abstract We study the symbol and the alphabet for two-loop NMHV amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills from the Q ¯ $$ \overline{Q} $$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use Q ¯ $$ \overline{Q} $$ equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.